### Nonlinear Dynamics and Statistics

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For values of A sufficiently small, the stationary state is monotonically achieved either from above or below. This represents the bifurcation point from which oscillations in levels of k occur for even higher values of A. Then, the author derives the following sufficient condition for the existence of growth cycles, that is:. Indeed, for a parameterizations satisfying such condition, the capital-labor ratio exhibits bounded oscillations perhaps after a period of growth.

In order to determine if such cycles can be chaotic, Day [ 6 ] shows the existence of a parameter set for the map such that the theorem in Li and Yorke [ 7 ] is satisfied. Figure 1. A Changes in the shape of the map, as A varies. Furthermore, some Inada like conditions are satisfied, in order to get an optimal bundle with strictly positive values for c and E.

The individual supplies inelastically a unit of labor only in the first period, earning a real wage w t , and optimally allocate w t between private savings s t 8 for the old age consumption and environmental improvement expenditures, m t. Then, the individual life-cycle budget constraints are given by:. The environmental quality evolves according to the following rule, staken from John and Pecchenino [ 9 ]:.

### Nonlinear Dynamical Systems

Firms maximize profits and then:. The representative agent maximizes his utility with respect to 19 and Then, the first order necessary and sufficient condition for the maximization problem reads as:. From the market clearing condition, the author gets the following relationship between capital and environment:. Proposition 1. That is, there exists a unique and asymptotically stable attracting positive steady state. It can be noticed that A does not play any role in the dynamic properties of the model: it only influences the level of E at the stationary state.

Figure 2. Occurrence of a period-doubling sequence as a 0 decreases. In the case of discrete time unidimensional models with overlapping generations, a variation of the model proposed in Zhang [ 10 ], with a more general production function, has been provided by Naimzada and Sodini [ 11 ]. Remaining close to the paradigm introduced by John and Pecchenino [ 9 ], the authors propose a model in which a population of individuals is characterized by a utility function depending on the stock of an environmental good, E t , and on the consumption of the private good, c t.

It is also assumed that E t is negatively affected by the consumption but it is improved by specific environmental expenditures. Differently from Zhang [ 10 ], Naimzada and Sodini [ 11 ] consider allocations in a decentralized economy. In this case, the choices of each agent between consumption and environmental expenditure generate externalities on the others.

A consequence of this assumption is that the environmental maintenance, because of the environment is considered as a public good, is characterized by the classic free-riding problem. In particular, the functional form described by the authors is the following:. Also in this case, Inada like conditions are also assumed to avoid corner solutions in the optimization problem.

On the production side, compared with [ 10 ], the model considers a more general specification, that is a CES-technology:. From the optimality conditions, equilibrium expressions of wage rate w t , and interest rate r t are derived as.

By assuming that agents are identical, the problem faced by the individual born at t is:. In order to analyse dynamics of the model, the authors, as in Zhang [ 10 ], introduce a constant elasticity of substitution between consumption and environment, defined as:.

### Non-linear Dynamics and Statistical Physics News

Hence, they characterize the intertemporal equilibrium conditions by means of a nonlinear difference equation in E t :. In the case i , the authors show that the map admits a unique positive fixed point and highlights that qualitative results similar to the ones in Zhang [ 10 ] are achievable. In the case ii , the map admits an odd number of fixed points and the ones with even index are unstable. Then, differently from i , multiple equilibria may exist.

An other interesting phenomenon shown by the authors is the following. Then, a change in A may be engine of a poverty trap see [ 12 ].

## Dynamical system - Wikipedia

Considering another configuration of parameters see the parameter set in Figure 3 , the authors provide another example in which a different sequence of dynamic phenomena occurs, as A varies. Figure 3.

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Moreover, there exists a region of parameters for which coexistence of attractors occurs see Figure 3. For larger values of A , the lower attractor dies and all the feasible trajectories are attracted by the remaining attractor that lives at the right of the repelling steady state E 2.

Figure 4 shows an interesting global phenomenon analyzed by the authors, for even higher values of A : the remaining attractor enlarges and invades the space before occupied by the other attractor. Finally, an uncommon phenomenon analyzed in the work is the following one. By investigating the second iterate G 2 , it can be noticed that an increase of A may lead to a fold bifurcation of G 2 inducing a stable 2-cycle and an unstable 2-cycle.

In this case, the fold bifurcation arises far from the fixed point that maintains its instability for the whole process see Figure 5. Figure 5. Figure reproduced with the permission of the authors. Source: Naimzada and Sodini [ 11 ]. Recently, by following some suggestions from empirical findings 9 , Caravaggio and Sodini [ 14 ] have proposed a model in which, by assuming an overlapping generation framework and discrete time, the environmental resource is not an argument in the agent's utility but is taken as an input in the production of a private good.

Moreover, differently from John and Pecchenino [ 9 ], the model on the one hand assumes that the economic activity damages the environment and that agents consider it only as an externality 10 ; on the other hand, it assumes that the government imposes a wage taxation with a fixed share devolved to the maintenance of a natural resource About the economic agent, at time t he supplies inelastically his time endowment and the earned wage w t is allocated between the consumption c t and the saving s t.

Thus, the utility function is given by:. Then, the agent has to face the following intertemporal budget constraint:. On the production side, the private good is produced through the following Cobb-Douglas technology:.

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From the known optimality conditions, the equilibrium expressions for wage and interest rate are derived as:. Regarding to the evolution of natural resource, a functional form similar to the one introduced in Smulders and Gradus [ 16 ] is adopted:. In order to analyse the dynamics of the system by studying the dynamic properties of the map N , some relationships between maps M and N are provided:. Figure 6. Figure reproduced with the permission of the publisher.

Source: Caravaggio and Sodini [ 14 ]. Authors also highlight the particular role played by the scaling parameter A. Figures 7A,B provide a numerical example of this occurrence in terms of k and E. Figure 7. A Evolution of capital accumulation over time when a shock in A occurs; B Evolution of the stock of environmental resources over time when a shock in A occurs. The red pointed vertical line represents the temporary exactly 10 iterations time series generated by A l.

In the case ii , Caravaggio and Sodini [ 14 ] underline two interesting results: 1 the noninvertibility and unimodality of the map may induce the birth of nonconnected basins of attraction see Figure 8 ;. Figure 8. Figure 9. Continuing with the literature on the discrete time OLG models, several works in which a different allocation problem for the agents is taken into account have been proposed, as in Antoci and Sodini [ 17 ] and in Antoci et al. In this context, differently from John and Pecchenino [ 9 ], where the investment in environmental defensive expenditures is internalized by agents in the allocation problem, economic agents are assumed to be unable of interven for the environmental maintenance.

These models therefore consider an environmental quality that is deteriorated by economic activity and individuals who defend themselves from such worsening by simply increasing the consumption of the private good, perceived as a substitute of the environmental one. As shown in Antoci and Sodini [ 17 ], since the production has a negative impact on the environment, self-protection choices induce further environmental degradation. Then, a self-enforcing mechanism may be observed according to which environmental depletion leads to an increase in the consumption of private good, which in turn generates further environmental degradation and so on.

These works show also that such self-reinforcing substitution mechanism may fuel undesirable growth paths according to which an increase in consumption of the private good is associated with a reduction in the individual well-being. Moreover, this mechanism may lead to the occurrence of several global phenomena and the individuals' expectations about the future evolution of the environmental quality can induce local and global indeterminacy.

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Before introducing the main features of the model in Antoci et al. Indeed, indeterminacy occurs if, for a given initial condition on the state variables, different values for the control variable can be chosen.